Green Technology and Patents in the Presence of Green

Consumers�

Corinne Langiniery and Amrita Ray Chaudhuriz

May 2019

Abstract

We develop a theoretical framework to investigate the impact of patent policies and emis-

sion taxes on green innovation that reduces the emission output ratio, and on the emission

level. In the absence of green consumers, the introduction of patents results in a paradox

whereby increasing emission tax beyond a certain threshold leads to a discrete increase in

the emission level, which may be avoided by reducing the patenting cost. In the presence

of green consumers, this paradox is restricted to an intermediate range of tax rates, and

at su¢ ciently high tax rates, reducing the patenting cost may increase the emission level.

Also, higher emission taxes increase green investment only if the fraction of green consumers

is su¢ ciently small, and the magnitude of this e¤ect decreases as this fraction increases.

Moreover, a stricter patentability requirement is only e¤ective at reducing emissions if the

fraction of green consumers is su¢ ciently small.

Keywords: Patent, Clean Technologies, Environmentally Friendly Consumers

JEL Classi�cation = O34, L13, Q50

�This work was carried out with the �nancial support of SAS grant, University of Alberta. We are grateful

to Agamani Chakrabarty for her research assistance. We would like to thank Stefan Ambec, Sophie Bernard,

Philippe Marcoul, and the participants at the seminar at Queen�s University, Smith School of Business, REES

department, University of Alberta, TECHNIS webinar, University of Crete, and of CAES 2017, IIOC 2018, CEA

2018 and WCERE 2018 for insightful comments. All remaining errors are ours.yDepartment of Economics, University of Alberta, Edmonton, Canada (langinie@ualberta.ca)zDepartment of Economics, University of Winnipeg, Canada, CentER & TILEC

(a.raychaudhuri@uwinnipeg.ca)

1 Introduction

Given increasing environmental concerns and the increasing environmental consciousness of cit-

izens globally, developing green technologies has become a key policy initiative of international

organizations such as the UN and the G8, and of national governments.1 Within this context,

we examine the interaction of patent policies and environmental regulation in enhancing inno-

vation in �green�, or less polluting, production technologies, and in reducing the emission of

pollutants. Studying patent policies for the development of green technologies requires a spe-

ci�c analysis due to potential interactions between knowledge and environmental externalities

caused by the innovators. We thus develop a theoretical framework to investigate the impact of

changing patentability requirements and patenting costs in conjunction with increasing emission

taxes, allowing for the presence of environmentally friendly consumers.

This paper builds upon existing results in the literatures on the development of green tech-

nologies, on patents, and on the environmentally friendly behaviour of consumers.

The literature on green technologies is rapidly evolving in response to global environmental

problems such as climate change. Climate experts propose that the increase in average global

temperature should be restricted to about 2�C to avoid the possibility of catastrophic damage,

a goal that may only be reached by developing and implementing �breakthrough� technolo-

gies that reduce emissions dramatically (Barrett, 2009; Galiana and Green, 2009). One stream

of the literature has evolved around the seminal work of Porter (1991) and Porter and van der

Linde (1995), referred to as the Porter Hypothesis, and examines whether the implementation of

stricter environmental regulations increases �rms�incentives to invest in green Research and De-

velopment (R&D). The empirical evidence surrounding the Porter Hypothesis is mixed (Ambec

et al., 2013). Our paper is more closely related to a second stream of the literature, which argues

1 In 2016, the Canadian federal government announced that it will invest $200 million annually to create

sector speci�c strategies to support the development of clean technologies and invest $100 million annually in

organizations that support clean technology �rms such as Sustainable Development Technology Canada. In the

U.S., the Department of Energy�s Loan Program O¢ ce has more than $40 billion in remaining loans to help

�nance innovative technologies that can reduce carbon emissions. In Canada and the U.S. respectively, about

2500 and 18500 patents for green technologies are issued annually.

2

that given the combination of environmental externalities and knowledge market failures facing

regulators, environmental policies are not su¢ cient to achieve the �rst best social outcome, and

need to be combined with policies addressing the relevant knowledge market failure (Carraro

and Siniscalco, 1994; Carraro and Soubeyran, 1996; Katsoulacos and Xepapadeas, 1996; Fischer

and Newell, 2008; Gerlagh et al., 2009; Acemoglu et al., 2012; Hepburn et al., 2018; Lehmann

and Söderholm, 2018; Popp, 2019).

Examining patent systems within this context is important due to the lack of consensus

among policy-makers and academics regarding the role of patents in promoting R&D in general

and green technologies in particular.2 On the one hand, international organizations advocate

royalty-free compulsory licensing of green technologies, excluding green technologies from patent-

ing, and even revoking existing patent rights on them (UNFCCC, 2009).3 On the other hand,

many countries actively lower the cost of obtaining patents for green innovations.

Our focus is on the role played by two di¤erent aspects of patent policies, patenting costs

and patentability requirements, and how they interact with emission taxes in fostering green

innovation.4 The cost associated with obtaining and implementing patents can be signi�cant

and comprises a number of components. First, the monetary fees associated with the application

process range on average between $5000-$15000. Second, there exists an opportunity cost in

terms of lost pro�ts while waiting for the patent to be granted. On average, the waiting time is

about three years and is frequently longer.5 Third, the potential litigation costs of enforcing a

patent may be large enough to deter small �rms from obtaining patents in several industries.6

2Patents address the problem due to the externality that results from imperfect appropriability of knowledge

by endowing innovators with property rights on their inventions. A patent confers its owner a temporary right

to exclude others from exploiting the innovation. In exchange for the exclusionary right, the patent holder must

disclose his innovation. For surveys of the patent literature, see Langinier and Moschini (2002), Rockett (2010),

Eckert and Langinier (2014).3Such provisions are also incorporated in the Agreement on Trade Related Aspects of Intellectual Property

Rights (TRIPS) (Derclaye, 2008; Rimmer, 2011).4Gerlagh et al., (2014) is another paper to examine patent policies for green technologies. It focuses on

analyzing the impact of changing the lifetime of patents issued to green innovations.5See Eckert and Langinier (2014).6According to the American Intellectual Property Law Association, the cost of an average patent lawsuit,

where $1 million to $25 million is at risk, is $1.6 million through the end of discovery and $2.8 million through

3

Policy makers use di¤erent means to reduce these patenting costs. One such method which

has been frequently used is fast-tracking, or expediting the review process, of green patent

applications. This is a key policy initiative undertaken by several countries including Australia,

Brazil, Canada, China, UK, U.S., Japan and Korea. Given that the time from application to

grant has been e¤ectively reduced by up to 75% for patents entering the fast track procedure

and that evidence shows that fast-tracking programs have accelerated the di¤usion of knowledge

in green technologies in the short run (Dechezleprêtre, 2013), such policies are likely to continue

and proliferate. In our model, we introduce a lump sum cost associated with obtaining and

implementing patents, and examine the impact of lowering this cost.

We also analyze the impact of changing another aspect of patent policy, that is, patentability

requirements. In order to be patentable an innovation must be su¢ ciently novel (not already in

the public domain), non-obvious (to a person with ordinary skills in the particular �eld), and

useful (to have at least one application). The relevant requirements vary across jurisdictions

and are currently stricter in the EU than in the U.S. (Eckert and Langinier, 2014). In the

spirit of Crampes and Langinier (2009), we model the patentability requirement as a minimum

investment threshold level that must be satis�ed. We then vary this investment threshold to

examine whether a stricter patentability requirement fosters more green innovation.7

We also contribute to the green innovation literature by incorporating environmentally

friendly consumers in our model. The increasing environmental consciousness of citizens globally

is re�ected in widely used eco-labeling schemes internationally.8 A few papers study optimal

�nal disposition.7Some studies have illustrated that strong Intellectual Property Rights may not necessarily enhance innovation

(Green and Scotchmer, 1995; Gallini, 2002; Bessen and Maskin, 2009). Even though in a static world (single

innovation), patents of appropriate scope can encourage innovations (Klemperer, 1990; Gilbert and Shapiro,

1990), this is no longer the case when the cumulative nature of innovation is accounted for. In the case of

cumulative innovations, strict patentability requirement may even discourage follow-on innovations (Scotchmer,

1991). The prospect of being imitated inhibits inventors in a static world but, in a dynamic world, imitators

can bene�t both the original inventor and society (Bessen and Maskin, 2009). In our paper, we abstract away

from these issues and present an alternative mechanism through which stronger patentability requirements a¤ect

innovation.8For example, in countries like Sweden about 50% of the market share for certain products consists of the

4

environmental policies in the presence of environmentally friendly consumers, but they do not

address green innovation (Arora and Gangopadhyay, 1995; Cremer and Thisse, 1999; Moraga-

Gonzalez and Padron-Fumero, 2002; Bansal and Gangopadhyay, 2003; Lombardini-Riipinen,

2005; Bansal, 2008). It is important to include environmentally friendly consumers since this

might modify the role of emission taxes in inducing innovation. This is because, as the product

becomes cleaner, the green conscious consumers demand more of it thereby mitigating the e¤ect

of emission taxes in reducing emissions.9 ;10 Our model is applicable to a wide variety of products,

the production processes of which are becoming cleaner. Consider for example, products such as

toys, furniture and packaging, the �greenness�of which may depend on the proportion of inputs

used in the production process that are recycled. Recent evidence suggests that the volume of

recycling has been increasing.11 Hence the importance of analyzing how behavioural responses

of environmentally friendly consumers to such cleaner production processes that generate the

�rebound-like e¤ect�in our model, a¤ect the functioning of policies such as emission taxes and

environmentally friendly variant. Green marketing is also frequently used to in�uence consumer behavior in

transportation and electricity markets (Kraftborsen, 2001).9This is reminiscent of the �rebound e¤ect�whereby as a product becomes more energy e¢ cient, demand for

the product increases (see Gillingham et al., 2016, for a survey). At the same time, we note that the channel

through which the demand increases in our model is di¤erent from the rebound e¤ect that is discussed in the

energy e¢ ciency literature. While in the latter, the increase in demand results from improved energy e¢ ciency

reducing the price of the good or the price of using the good, the �rebound-like e¤ect� in our model is driven by

environmentally friendly behaviour of consumers.10Another di¤erence between the rebound e¤ect that is discussed in the energy e¢ ciency literature, and the

�rebound-like e¤ect� in our model is that while the former is applicable to scenarios where innovation leads to a

cleaner consumption process, our model is applicable to scenarios where innovation leads to a cleaner production

process associated with a given product. For example, replacing a gasoline powered vehicle with a hybrid car

reduces the pollution externality related to the consumption of the good (in this case, the pollution arising from

driving). This paper, on the other hand, focuses on the case where the production process of the good causes less

pollution. By focusing on the production related pollution externalities, rather than consumption related ones,

we abstract away from cases where increasing demand caused by the availability of a cleaner product leads to less

pollution, as when a gasoline powered vehicle is replaced by a consumer with a hybrid car.11See Figures A1 to A2 in Appendix A which illustrate the increasing volume of recycling over time. For

example, over the period 2005-2016, the volume of recycling in Europe increased by 34%.

5

patents for green innovation.12

We model the market for a product, the production of which causes pollution. We assume

that the implementation of a cleaner technology results in a lower emission per unit of output

ratio (similar to, for example, Benchekroun and Ray Chaudhuri, 2014, 2015). Moreover, similar

to Bansal and Gangopadhyay (2003) and Ibañez and Grolleau (2008), we assume that the

product is vertically di¤erentiated in terms of its emission-output ratio with green conscious

consumers preferring products with lower emission-output ratios. On the demand side, we allow

consumers to be heterogeneous in terms of their degree of environmental friendliness. Our

framework has two stages where, in the �rst stage, an incumbent monopolist decides its level of

investment in R&D, and in the second stage, it chooses the price of its product. The monopolist

faces potential entry if it does not innovate. We assume that investment by the �rm reduces the

emission-output ratio. If the innovation is patented, the �rm e¤ectively behaves as a monopolist

when setting its price in the second stage. If the innovation does not satisfy the patentability

requirement, the �rm cannot patent it and faces Bertrand competition in the second stage since

entry occurs and rival �rms are assumed to have free access to the new technology.

Within this setting, our main �ndings are two-fold.

The �rst set of results shows that, in the absence of green consumers, the introduction of

patents may result in a paradoxical result whereby increasing the emission tax beyond a certain

threshold makes the innovation unpro�table, and thereby leads to a discrete increase in the

emission level. By introducing a �xed cost of obtaining patents, we obtain that increasing

the emission tax beyond a certain threshold makes the innovation unpro�table. At this tax

threshold, without the innovation, the emission level increases discretely to the competitive

emission level, causing the paradox. Reducing the patenting cost, e.g., by fast-tracking green

patents, decreases the likelihood that this paradox occurs. In the presence of green consumers,

this paradox occurs only within an intermediate range of tax rates. This is because, at very

12The rebound e¤ect as captured in the existing energy e¢ ciency literature is an important phenomenon.

Although the �rebound-like e¤ect� that we capture in our model has not yet received as much attention in the

existing literature as the traditional rebound e¤ect, given its potential empirical relevance, it is useful to examine

its policy implications.

6

high tax rates, the emission level in the competitive equilibrium that occurs in the absence

of innovation, is lower than that with innovation. If innovation occurred, demand from green

consumers would increase thereby increasing the emission level above that in the competitive

case without innovation. Thus, at very high tax rates, a lower emission level is reached despite

a tax increase that makes the innovation unpro�table. It follows that, at su¢ ciently high tax

rates, reducing patenting costs in order to induce innovation ends up increasing the emission

level in the presence of green consumers. The lowest level of emissions, in the presence of green

consumers, is therefore reached with very high emission tax rates combined with high patenting

costs.

The second set of results shows that the traditional policy tools of increasing green investment

and thereby reducing emissions through stricter emission taxes and patentability requirements

may become less e¤ective as society becomes more environmentally friendly, with consumers

rewarding marginal reductions in emission-ouput ratios of production processes. More speci�-

cally, while for a su¢ ciently small fraction of green consumers we retrieve the expected result

found in much of the literature surrounding the Porter hypothesis that a higher emission tax

increases green investment, this result is reversed if the fraction of green consumers rises to a

level such that investment in green technologies results in more emissions. These results are

driven by the �rebound-like� e¤ect introduced by the green consumers. We also show that a

stricter patentability requirement is only e¤ective at reducing emissions as long as the fraction

of green consumers is su¢ ciently small. Finally, we show that as long as the fraction of green

conscious consumers is su¢ ciently low, the �rm underinvests relative to the socially optimal

level for a su¢ ciently low emission tax and overinvests for a su¢ ciently high emission tax. How-

ever, the gap between the socially e¢ cient and privately optimal levels of investment steadily

reduces as the fraction of green conscious consumers increases, until this result is reversed when

this fraction becomes su¢ ciently large. Thus, further research seems warranted regarding the

policies to reduce emissions through green innovation and also regarding the type of information

to distribute to consumers.

We extend our analysis to a duopoly setting, where a single �rm decides to invest in a cleaner

technology and patent its innovation in order to attract green consumers. Our results in the

7

duopoly setting remain qualitatively similar to those derived in the monopoly setting.

The paper is organized as follows. In Section 2, we present our model. In Section 3, we

present the benchmark case without green consumers. In Section 4, we derive the equilibrium

and policy implications for the case with green consumers. Section 5 presents an extension of

the model with a duopoly case. Section 6 presents our concluding remarks. All proofs have been

relegated to Appendices A and B.

2 The Model

We consider a two-stage model in which a �rm sells a �nal good to consumers in a competitive

market, the production of which is polluting and has a marginal cost, c. The emission of the

pollutant generated per unit of production is given by:

� e

q; (1)

where e denotes emission and q denotes output. The �rm can invest IG to reduce the emission-

output ratio, . Thus, is a function of IG; where (IG) is such that 0 (IG) < 0; 00 (IG) > 0;

(0) = H and limIG!1

(IG) = L > 0. The higher is IG, the greener the product. For notational

convenience, henceforth we do not mention the argument of the function .

2.1 The demand side

The demand side consists of a continuum of N consumers. Each of them buys either 0 or 1 unit

of the good. There exists a fraction � of �green conscious�consumers, whose utility is increasing

in the �greenness�of the product, that is, decreasing in , and a fraction (1� �) of �non-green

conscious�consumers, whose utility is independent of the greenness of the product.

Let G denote the degree of environmental friendliness of a consumer, with G being uniformly

distributed over the interval�G;G

�with G > 0. We assume that consumers can observe how

green a product is. Within this context, this is equivalent to assuming that consumers can

observe .13 We normalize N such that N = 1; and assume that G � G = 1. Let P (e) denote13This is a relevant scenario to consider in the presence of e¤ective eco-labeling programs.

8

the pollution damage to each consumer, which is a function of total emissions, e. Following

Ibañez and Grolleau (2008), we assume that the pollution level generated by total production

is exogenous to each consumer, regardless of his consumption level. Let p denote the product

price.

A green conscious consumer has the following utility function:

UG =

8<: v �G � p� P (e) from buying the product

�P (e) from not buying(2)

The term �G in (2) re�ects that the greener the product, that is, the lower is , the better o¤

the green conscious consumer. Also, v represents the gross utility of consuming one unit of the

good. A green conscious consumer does not buy the product if v �G � p < 0:

A non-green conscious consumer has the following utility function:

UNG =

8<: v � p� P (e) from buying the product

�P (e) from not buying(3)

Henceforth, we assume that P (e) = e.

From (3), it follows that a non-green conscious consumer buys the good as long as v � p:

Therefore, if v < p, neither non-green conscious consumers nor green conscious consumers buy

the good. If p = v, only non-green conscious consumers buy the good. When p < v, non-

green conscious consumers always buy the good whereas a green conscious consumer with a

degree of environmental friendliness G buys the good as long as v � G � p � 0. There exists

a green conscious consumer eG who is indi¤erent between buying the good or not such that

v � eG� p� P (e) = �P (e) or eG = (v � p)= . As long as G < eG < G; some, but not all, greenconscious consumers buy the good. Thus, it follows that the demand function, D (p) ; is given

by:

D (p) =

8>>>>>>><>>>>>>>:

1 if p � v � G

�(v�p �G) + (1� �) if v � G < p < v � G

(1� �) if v � G � p � v

0 if p > v

(4)

9

2.2 Policy Tools

We consider a combination of policy tools. On the R&D side, we model a patenting policy for

green innovations. On the environmental side, we assume that the �rm must pay a tax, � ; per

unit of emission: Thus, the tax bill faced by the �rm is given by � D(p); where, by (1), D(p)

represents the emissions generated by the �rm.

The patent policy is such that the �rm must discover a su¢ ciently novel innovation to be able

to obtain a patent. We assume that novelty of the innovation is increasing in the investment level.

Recall that 0 (IG) < 0, such that a higher investment level reduces the emission-output ratio.

Thus, there exists a threshold P ; corresponding to investment level IPG; where L < P < H ;

such that a patent is only granted if an innovation reduces below P :14 Therefore, the �rm

must invest IG � IPG in order to ensure that � P . We consider weak and strong patentability

requirements, representing di¤erent levels of IPG; as de�ned in Section 4.3 by De�nition 1.

In order to obtain a patent, the �rm must also incur an exogenously given cost CPG; which is

broadly de�ned to include a monetary fee payable by the �rm to the patent o¢ ce, the opportunity

cost in terms of lost pro�ts incurred while waiting for the patent to be granted, as well potential

litigation costs for enforcing the patent. There are several ways in which policy makers may

reduce CPG; including by implementing a fast-track patent system for green technologies that

reduces the patent application processing time for green innovations.

Once a patent is granted and the �rm starts producing at a lower emission output ratio, this

lower value of � P becomes the new technology standard for the industry that is enforced

by regulators. That is, we implicitly assume that no rival �rm can enter the industry if its

production results in a higher : If the innovation does not satisfy the relevant patentability

requirements, the �rm cannot patent it, and faces Bertrand competition in the second stage.

We note that such technology-based standards are widely used for environmental regulation.15

14We implicitly assume that the novelty of the innovation is assessed by an experienced patent examiner who

is able to evaluate whether the innovation meets the patentability requirement.15For example, in Germany technology standards were used to reduce sulphur-dioxide emissions. In Canada,

under the Canadian Enviromental Protection Act, technology standards apply to a number of industries including

the energy sector, pulp and paper mills, and mineral smelters.

10

Moreover, technology standards are widely used as a trade barrier to keep foreign competition

out of domestic markets.16 In order to implement technology standards, the regulator prescribes

certain technologies, design standards, engineering standards or input standards which require

potential polluters to use inputs and production processes meeting speci�c conditions. Typically,

technology standards specify that polluters use the �best available technology�(BAT),17 imply-

ing that, subsequent to an innovation which improves the best available technology in a given

industry regulated by technology standards, �rms with dirtier technologies would be unable to

enter the industry.

2.3 Timing

The timing of the game is as follows. There are two stages. In the �rst stage of the game, the

�rm decides the level of investment in the green technology, IG.18 Once an innovation has been

discovered, the �rm decides whether to patent it.19 In the second stage, the �rm chooses the

price of the product it o¤ers, p:

We solve for the equilibrium investment level and price through backward induction. For a

given level of investment, we �rst determine the pricing strategy of the �rm. Then, we determine

the level of investment at the equilibrium.

As a benchmark case, we �rst consider the scenario where there exist only non-green conscious

consumers (i.e., � = 0) in Section 3. Next, we enrich our analysis by considering that both green

and non-green conscious consumers exist (i.e., 0 < � � 1), in Section 4.16For further details, please refer to: https://www.international.gc.ca/trade-agreements-accords-

commerciaux/topics-domaines/goods-produits/barriers.aspx?lang=eng17See, for example, Field and Olewiler (2011).18An alternate interpretation of the investment level, IG, in our model is that it represents the cost incurred

by the �rm for obtaining the innovation.19We make the simplifying assumption that an innovation will be discovered with probability 1. If we alterna-

tively assumed that an innovation will be discovered with an exogenously given probability less than 1, this would

make the notation and analysis more cumbersome without generating any new insights.

11

3 Benchmark Case: Non-Green Conscious Consumers

In this section, we present the scenario without any green consumers, that is, � = 0: In the

second stage, we have monopoly pricing, whereby the pro�t maximizing price is given by p = v:

From (4) ; it follows that, for � = 0; the demand is given by D = 1. Thus, the emission level is

given by e = (IG) ; and the pro�t is given by (v � c� � ). Recall that c denotes the marginal

cost of production.

In the �rst stage, the �rm�s optimization problem is given by:�MaxIG

[v � c� � (IG)� IG]

s.t. � P � (IPG)

Let ING be the solution of the unconstrained program (i.e., �� 0(ING) = 1); the �rm therefore

chooses the optimal investment INGG = maxfING; IPGg. As long as the unconstrained investment

is large enough (ING > IPG) so that the �rm optimally chooses to reduce its emission-output

ratio su¢ ciently, it is not constrained by the patentability requirement. However, if ING < IPG,

in order to patent its innovation, the �rm has to increase its investment to IPG, which is beyond

its unconstrained optimal investment.

The unconstrained optimization investment level ING is increasing with the tax � so that

the emission level is decreasing. However, for very low values of � (in the extreme case if � = 0),

the �rm has very little (no) incentive to invest, so that 0 � ING < IPG is always satis�ed.

Consider two threshold levels of emission tax, �̂1 and �̂2; such that P = e�ING (�̂1)

�and

CPG = v� c� �̂2 �ING (�̂2)

�� ING (�̂2) : Our �rst set of �ndings concerning the emission level

is summarized in Proposition 1.

Proposition 1 The emission level, with � = 0

(i) is given by P for � 2 [0; �̂1);

(ii) decreases due to marginal increases in the tax rate at any � 2 [�̂1; �̂2);

(iii) increases discretely due to an increase in the tax rate from any � < �̂2 to any � > �̂2:

The �ndings listed in Proposition 1 are illustrated in Figure 1 below (see Appendix A for

the explanation regarding the shapes of the functions in all the �gures).

12

Figure 1: Pro�t maximizing emission levels when � = 0

For a su¢ ciently low tax rate, that is, � 2 [0; �̂1); the �rm invests the minimum amount

required to obtain a patent, that is, IPG. For � 2 [�̂1; �̂2), the �rm has an incentive to invest

more than this minimum level since �ING

�< P . We note that the �rm only invests as long

as

v � c� � �ING

�� ING � CPG > 0; (5)

which is not satis�ed for � > �̂2: Thus, for � > �̂2, the �rm does not invest and Bertrand

competition results in emission level given by ec = H :

Proposition 1 highlights the �rst contribution of our paper, which is to show that the in-

troduction of patents may result in a paradoxical result whereby increasing the emission tax

beyond a certain threshold leads to a discrete increase in the emission level. Moreover, it follows

from (5) that the higher the patenting cost, CPG; the greater the range of taxes for which the

paradox occurs.

Corollary 1 The lower the patenting cost, CPG; the less likely that the emission level increases

as emission tax increases.

A direct policy implication of Corollary 1 is that if CPG is reduced by any means, such as

fast-tracking green patents, the less likely that this paradox occurs and the more e¤ective are

emission taxes at inducing green innovation. This follows from the fact that the threshold �̂2 is

13

decreasing in CPG; by de�nition.20

In the following section, we examine whether the results summarized in Proposition 1 and

Corollary 1 carry over to the scenario with green consumers.

4 The Equilibrium with Green Consumers

In this section, we continue our analysis with the general model introduced in Section 2 with

green consumers, that is, 0 < � � 1:

4.1 Second Stage: Monopoly Pricing

We begin with the second stage, and analyze the pricing strategy of the �rm. Assuming that

the innovation has been patented, in the second period the �rm solves the following:

maxp� � (p� c)D(p)� � D(p); (6)

where the demand is given by (4). Due to the discontinuities in the demand function, depending

on the chosen price, the �rm will face either all green and non-green consumers, some green

consumers and all non-green consumers, or only non-green consumers. The only relevant cases

for our analysis are those where non-green consumers and some green consumers consume, and

only non-green consumers buy the good. Therefore, we will only consider a constellation of

parameters such that the cases where demand is inelastic at 1 or at 0 are ruled out.

If only non-green consumers buy the good, the demand is (1� �) such that the �rm sets its

price at v and, therefore, the pro�t is given by:

�NG = (v � c� � )(1� �); (7)

where the emission level is thus e = (1� �).

If some green conscious consumers and all non-green conscious consumers buy the good, the

pro�t-maximizing price is given by:

pm(IG) =1

2(v + c� (G� �)) + 1� �

2� ; (8)

20Recall that CPG = v � c� �̂2 �ING (�̂2)

�� IPG: Since IPG is constant and v � c� �

�ING

�is decreasing

in � ; we have that an increase in CPG must correspond to a decrease in �̂2:

14

and the second order condition is satis�ed since @2�=@p2 = �2�= < 0.

In order to get demand from both types of consumers, we need to ensure that pm(IG) < v� G

or, equivalently, that � > �, where

� � +v�c� (G+�) > 0:

We show that � < 1 if < (v� c)=(G+ �). In order for this condition to be satis�ed for any ,

we assume that

H <v � cG+ �

; (A1)

so that for any < H , we have � < 1.

From (8), it follows that the demand is given by:

D =�

2 [v � c� (G+ �)] + 1

2(1� �): (9)

For values of � > �, the demand function, evaluated at the price pm(IG); is decreasing in the

marginal cost, c, and is increasing in the valuation, v: Since @D=@ = �� (v � c) =2 2 < 0; we

have that the demand is increasing in IG. As the investment in the green technology increases,

the demand increases as the product becomes greener.

To ensure that some but not all green conscious consumers buy the product (and rule out

the case of inelastic demand 1), we further assume that

L >v � cG+ �

: (A2)

Therefore, assumptions (A1) and (A2) ensure that the extreme cases of D = 0 and D = 1 are

avoided, since these cases would lead to discontinuities.21

By substituting (8) and (9) into (6) ; when both non-green and green conscious consumers

buy, we obtain the net pro�t of the �rm in the second stage as the following:

�m =�

4 [v � c� (G+ �) + 1��

� ]2: (10)

21More speci�cally, the condition H < (v�c)=(G+�) implies that the least environmentally friendly consumer

has a positive demand for the dirtiest good ensuring that D 6= 0 for � = 1, and together with (A2) ensures that

D 6= 0 for all � > �: Moreover, the condition L > (v� c)=(G+ �) implies that the most environmentally friendly

consumer does not buy the cleanest good ensuring that D 6= 1 for � = 1: Also, L > (v� c)=(G+ �) is a su¢ cient

condition to ensure that eG < G for all � > �: Thus, Assumptions (A1) and (A2) together ensure that G < eG < G:15

Even if both types of consumers are willing to buy the good (i.e., for � > �), the �rm can

either choose to serve only non-green consumers and set its price at v; or to serve all types of

consumers by setting the price (8). The �rm will choose to serve only non-green consumers if

�NG > �m where �NG is de�ned by (7) and �m by (10). This is equivalent to having � < �1

where

�1 �

+v�c+ (G��)�2(v�c� �)12 ( G)

12. (11)

See Appendix A for details of the derivation of �1. It is straightforward to show that �1 > �

as long as (A1) is satis�ed: Therefore, if the fraction of green consumers is small (� < �), only

non-green consumers demand the product. However, even if green consumers are willing to buy

the good, the �rm might set its price at v such that only non-green consumers will purchase.

Thus, for � 2 [�; �1], only non-green consumers buy as the price is too high for green consumers

to buy. For � 2 [�1; 1], both types of consumers buy the good at the price given by (8).

To summarize, when � 2 [�1; 1], the �rm chooses the monopoly price (8) in order to serve

all types of consumers. When � 2 [0; �1], only non-green consumers buy the good at price v.

Thus, for � 2 [�1; 1]; from (9) it follows that for any IG; the emission level of the �rm is

given by:

em(IG) =�

2[v � c� (G+ �)] +

2(1� �): (12)

Let

�G �1

G+ � + 1:

The following Lemma follows directly from (12) (the proof is provided in Appendix A).

Lemma 1 The emission level, em(IG);

(i) is decreasing in the emission tax, � ; and

(ii) is decreasing in IG for � 2 (�1; �G]; and increasing in IG for � 2 (�G; 1]:

In the absence of many green-conscious consumers, i.e., � 2 (�1; �G]; we obtain the standard

result that an increase in investment in green technology results in less emissions. However, the

rate of decrease of emissions steadily reduces as � increases (since @2em(IG)@IG@�

= �12 (G+ � + 1)

d dIG

>

16

0) until it becomes positive for � > �G: Thus, Lemma 1(ii) implies that private investment by

�rms in green technologies may lead to more emissions if the fraction of green conscious con-

sumers is su¢ ciently large, i.e., � > �G: This is because, from (9), we have that demand is

increasing in IG since as the product becomes cleaner, the environmentally friendly consumers

demand more of it.22 Moreover, Lemma 1(i) states that regardless of the impact of IG on the

emission level, an increase in the emission tax rate decreases the emission level. Lemma 1(i)

and (ii) together imply that when �rms invest in green technologies in the presence of green

consumers who increase their demand for cleaner products, it becomes necessary to implement

environmental regulation, such as an emission tax to ensure a reduction in the emission level.

Figure 2 illustrates, in more detail, the impact of increasing investment in green technology

on emissions.

Figure 26

-

1

H

Area I

Area III

@em(IG)@IG

< 0

�

1G+1+�

L

�1

@em(IG)@IG

< 0

@em(IG)@IG

> 0

Area II

In Figure 2, in Areas I and II, we have @em(IG)@IG

< 0; and in Area III, we have @em(IG)@IG

> 0:

Area I represents the combinations of and � for which the monopolist chooses to serve only

22This is reminiscent of the rebound e¤ect, as explained earlier in Footnote 9.

17

non-green conscious consumers by setting p = v. This occurs for � 2 (0; �1). In fact, for

� 2 (0; �) only non-green conscious consumers demand the good, while for � 2 (�; �1), some

green conscious consumers are willing to buy, but the �rm chooses to set a price that is too high

for them to buy. In Area I, the emission level is given by:

em(IG) = (1� �);

and thus,

@em(IG)@IG

= 0(1� �) < 0:

For � > �1; the monopolist chooses pm and serves both green and non-green consumers

(Areas II and III in Figure 2). The emission level in Areas II and III is given by (12) : Area II

represents the case where � 2 (�1; �G) such that @em(IG)@IG

< 0; as per Lemma 1, and Area III

represents the case where � > �G such that@em(IG)@IG

> 0; as per Lemma 1.

4.2 First Stage: Pro�t-Maximizing Investment

In the �rst stage of the game, the �rm chooses the investment level in green technology IG that

solves the following problem: � MaxIG

�m(IG)� IGs.t. (IG) � P � (IPG)

where the pro�t �m(IG) is de�ned by (10) for � 2 (�1; 1].23 Let ImG be the solution of the

unconstrained optimization problem (i.e., ImG = argmax�m(IG)�IG) such that it is the solution

of the following �rst order condition:

� 0(IG)[�4 ((v�c (IG)

)2 � (G+ �)2)| {z }(i)

+ (1��)2 (G+ �)� 1

4(1��)2�| {z }

(ii)

] = 1: (13)

We note that the left-hand side of equation (13) is positive such that (13) is satis�ed as long

as Assumptions (A1) and (A2) hold (see Appendix A for details). The �rst term (i) of (13)

captures the change in demand of green consumers as investment changes marginally, whereas

the second term (ii) captures the change in demand of non-green consumers. The second term

23The case where � 2 [0; �1) is similar to the benchmark case with only non-green consumers.

18

can push the investment level down since it can be negative. To obtain an interior maximum

solution, we further assume that

00(IG) > �� (v�c)2

2 ( 0(IG) (IG)

)3; (A3)

such that the second order condition is satis�ed (see Appendix A for details).

The solution of the constrained problem is thus Im � MaxfImG ; IPGg. Indeed, as long as

the constraint is always satis�ed, we have ImG > IPG and thus the �rm chooses its optimal

investment level. When the constraint binds, the �rm must invest IPG > ImG .

4.3 Policy Implications

Having solved for the equilibrium of the model, we now turn to analyzing the e¤ect of changes

in emission tax and patent policies. In order to do so, we �rst de�ne �weak� and �strong�

patentability requirements.

De�nition 1 A �weak� patentability requirement is de�ned to be IPG � ImG for � = 0; and a

�strong�patentability requirement is de�ned to be IPG > ImG for all � � 0:

By De�nition 1, a strong patentability requirement means that no matter what the emission

tax level, the unconstrained optimal investment level of the monopolist, ImG ; is always below the

required investment level to obtain a patent, IPG. A weak patentability requirement means that

in the absence of any emission tax, the monopolist naturally invests more than the minimum

required to obtain a patent. However, ImG is decreasing with the tax level,24 so that as the tax

increases we may have IPG � ImG .

In what follows, we consider that the fraction of green conscious consumers is such that

� > �1. When the patentability requirement is weak, we begin by focusing on the case where

the �rm chooses to invest ImG in the �rst stage, patents its green innovation and sets the price

p(ImG ) in the second stage as long as �m(ImG )� ImG � CPG > 0.25

24See the section on comparative statics in Appendix 1 for further details. A summary of the results is provided

in Table 1 in Appendix A.25See Appendix 1 for comparative statics with respect to c and v:

19

Lemma 2 For a weak patentability requirement, investment in green technology, ImG ;

(i) is increasing in the emission tax, � ; and in G; for � 2 (�1; �G];

(ii) is decreasing in the emission tax, � ; and in G; for � 2 (�G; 1]:

Lemma 2 follows directly from (13) (see Appendix A for details of the calculations): In the

absence of many green conscious consumers, i.e., � 2 (�1; �G]; we obtain the standard result that

an increase in emission tax induces greater investment in green technology. However, the rate of

increase of investment steadily reduces as � increases, until it becomes negative for � > �G: This

is because by Lemma 1; an increase in IG would be accompanied by an increase in emissions

for � > �G, and thereby, an increase in the tax bill facing the �rm. Thus, for � > �G; the �rm

chooses to decrease green investment when faced with a higher emission tax. Changes in G play

a similar role to changes in � in our model.

The intuition behind Lemma 2 is that while trying to lower the tax bill by increasing inno-

vation, the �rm faces a trade-o¤ between lower emission per unit of good and higher demand.

The former e¤ect outweighs the latter when the fraction of green consumers is not too high, but

the opposite is true when the fraction of green consumers becomes too large.

For � 2 (�G; 1]; an implication of Lemma 2 is that ImG keeps falling as � increases until we

have IPG > ImG : Let ~� denote that level of the tax rate where ImG = IPG: For � > ~� ; the �rm must

invest more than ImG in order to satisfy the patentability requirement. That is, the investment

level is given by Im �MaxfImG ; IPGg. Thus, the equilibrium price is given by p = pm(Im): We

summarize this �nding in the following Lemma.

Lemma 3 If CPG < �m(IG)� IG, the �rm

(i) invests Im � maxfIPG; ImG g in the �rst period, patents its innovation and

(ii) chooses the price p = pm(Im) as de�ned by (8) in the second period.

The quantity sold and the emission level in equilibrium are given by:

qm(Im) =�

2 (Im)[v � c� (Im) (G+ �)] + 1

2(1� �);

20

and

em(Im) =�

2[v � c� (Im) (G+ �)] + 1

2(1� �) (Im):

If IPG is larger, that is, the patentability requirement is strong as per De�nition 1, the �rm

must invest IPG for any � � 0 in order to satisfy the patentability requirement and will also set

a higher price pm(IPG) > pm(ImG ). It will invest as long as �m(IPG)� IPG � CPG > 0.

Thus far, we have presented the case where the �rm invests in the green technology and

patents it. However, the decision regarding whether to invest depends on the patenting cost,

CPG.

If the patenting cost is relatively small (CPG < �m(IPG)� IPG), the �rm always invests, no

matter how stringent the patent policy. If the patenting cost is very large, (CPG > �m(ImG )�ImG ),

the �rm never invests. For intermediate values of the patenting cost (�m(IPG)� IPG < CPG <

�m(ImG ) � ImG ), a too stringent patentability requirement discourages the �rm from investing,

whereas a less strict patentability requirement induces the �rm to invest. We summarize these

�ndings in the following Lemma.

Lemma 4 The �rm invests in the green technology

(i) if the patenting cost is small (CPG < �m(IPG)� IPG), or

(ii) if the patenting cost is higher (�m(IPG)�IPG < CPG < �m(ImG )�ImG ), but the patentability

requirement is not strong (IPG � ImG ).

If the conditions in terms of CPG; as per Lemma 4; are not satis�ed, the �rm does not invest,

such that Bertrand competition occurs in the second stage of the game with = H . In this case,

the price is given by pc = c+ H� , the demand is given byD(pc) = �(v�c� H(G+�))= H+(1��)

and the emission level is given by:

ec = �(v � c� H(G+ �)) + H(1� �): (14)

Lemmas 2, 3 and 4 are summarized in Figures 3 and 4 below. Using these Figures, we examine

whether the paradox that occurs in the benchmark case without green consumers carries over

21

to the case with green consumers: �rst under a weak patentability requirement (Figure 3), and

then under a strong patentability requirement (Figure 4).

Figure 3 illustrates the equilibrium emission level as a function of � under a weak patentability

requirement if � > �Gj�=0(= 1=(G+ 1)) and H < (v � c)=2G (see Appendix A for details).

Let �1 denote the threshold level of � such that em (ImG ) = em (IPG) : Let �2 denote the

threshold level of � such that CPG = �m(IPG)j�=�2 � IPG: Let �3 denote the threshold level

of � such that em (IPG) = ec: Let b� denote the threshold level of � such that ec = 0: The top

(red) function represents ec, the curved (blue) one represents em(Im) when Im = ImG and the

lower (green) one represents em(Im) when Im = IPG. The pro�t maximizing emission level is

represented in bold.26

Figure 3: pro�t maximizing emission levels under weak patentability requirement

From Figure 3, it follows that, under a weak patentability requirement regime, for � < �1; a

su¢ ciently small increase in � (a stronger environmental policy), reduces the emission level in

equilibrium. A larger increase in � to beyond �1 pushes the pro�t-maximizing investment level

down to IPG in which case, in order to be able to patent the �rm cannot reduce its investment

anymore and, thus, must invest at IPG. At this investment level, as � increases further, the

emission level decreases but at a slower rate. At even higher values of � ; i.e., beyond �2; the

26Figure 3 illustrates the case for � 2 (�G; 1]: For � 2 (�1; �G]; at � = 0 we have em (ImG ) < em (IPG) ; since

ImG > IPG and we are in Area II of Figure 2 whereby @em(IG)@IG

< 0: As � increases, �G decreases such that we

enter Area III of Figure 2 whereby @em(IG)@IG

> 0: At the value of � where this happens, em (ImG ) crosses em (IPG) ;

and for higher values of � we have em (ImG ) > em (IPG) : This does not qualitatively change our results.

22

�rm decides not to patent, and thus not to invest, in which case the emission level goes up to

the competitive level, ec as shown in Figure 3. At a still higher tax rate, i.e., beyond �3; ec

falls below the emission level that would be reached if the innovation had occurred, em (IPG).

Thus, unlike in the benchmark case with only non-green consumers the paradoxical result occurs

only for an intermediate range of tax rates, � 2 (�2; �3). Holding constant CPG; the paradox

disappears by increasing tax rates beyond �3 which reduces the emission level to ec, despite the

lack of innovation. At tax rates above �3, we have em (IPG) > ec for the following reason. While

in the competitive equilibrium, there is no innovation, if IPG were invested, demand from the

green consumers would be higher, pushing up the emission level.

Proposition 2 The emission level, in the presence of green consumers,

(i) decreases due to marginal increases in the tax rate at any � 2 [0; �2[ and � 2]�2;b� ];(ii) increases discretely due to a marginal increase in the tax rate at �2:

Proposition 2 generalizes the �ndings stated in Proposition 1 to the case with green consumers

(see Appendix A for proof) with a caveat for very high tax rates. Comparing Proposition 1 and

Proposition 2; it follows that for large emission tax levels, � > �3, an increase in the emission

tax level is only e¤ective at reducing the emission level in the presence of green consumers.

We note that the threshold �2 is decreasing in CPG; by de�nition:27 This leads to the following

Corollary.

Corollary 2 A su¢ cient reduction in the cost of patenting, CPG;

(i) decreases the emission level for an intermediate range of tax rates, � 2 (�2; �3);

(ii) increases the emission level for very high tax rates, � > �3:

Corollary 2 determines the conditions under which reducing patenting costs (e.g., by fast-

tracking green patents) helps to reduce emission levels, and when it does not. As per Corollary

27Recall that CPG = �m(IPG)j�=�2 � IPG: Since IPG is constant and �m(IPG) is decreasing in � ; we have

that an increase in CPG must correspond to a decrease in �2:

23

2(i), since the increase in emission occurs due to �m(ImG )�ImG falling below CPG after an increase

in � beyond �2; this adverse impact could be avoided by decreasing CPG; as in the benchmark

case without green consumers. However, in the presence of green consumers and for tax rates

higher than �3, as per Proposition 2(iii), decreasing CPG increases the emission level from ec

(which would occur without innovation if CPG were su¢ ciently high), to em (IPG) ; as shown in

Figure 3.

The same results carry over to the case with a strong patentability requirement such that

IPG > ImG for all � � 0, as per De�nition 1. As shown in Figure 4, the emission level decreases as

� increases, and the same mechanism applies as described above, except that the �rm is always

constrained to invest IPG. Note that the thresholds �2; �3 and b� are the same as in Figure3. In Figure 4, the top (red) function represents ec, the curved (dashed blue) one represents

em(Im) when Im = ImG and the higher (green) one represents em(Im) when Im = IPG. The

pro�t maximizing emission level is represented in bold. The paradoxical result occurs for the

intermediate range of tax rates, � 2 (�2; �3).28

Figure 4: Pro�t maximizing emission levels with strong patent requirement

Next, we compare strict and weak patentability requirements in terms of their impact on

emission levels.28Figure 4 illustrates the case for � 2 (�G; 1]: For � 2 (�1; �G]; we have em (ImG ) > em (IPG) ; since ImG < IPG

and we are in Area II of Figure 2 whereby @em(IG)@IG

< 0: As � increases, �G decreases such that we enter Area III

of Figure 2 whereby @em(IG)@IG

> 0: At the value of � where this happens, em (ImG ) crosses em (IPG) ; and for higher

values of � we have em (ImG ) < em (IPG) : This does not qualitatively change our results.

24

Proposition 3 The emission level is

(i) higher under a weak patentability requirement than under a strong patentability requirement

for � 2 (�1; �G];

(ii) lower under a weak patentability requirement than under a strong patentability requirement

for � 2 (�G; 1]:

Proposition 3 follows directly from Lemma 1; and has direct and important policy implica-

tions. When the fraction of green conscious consumers is su¢ ciently small, i.e., � 2 (�1; �G];

moving from a weak to a strong patentability requirement for green innovation, ceteris paribus,

decreases the emission level. This corresponds to Areas I and II in Figure 2. However, when

the fraction of green conscious consumers is su¢ ciently large, i.e., � 2 (�G; 1]; moving from a

weak to a strong patentability requirement for green innovation, ceteris paribus, increases the

emission level, corresponding to Area III in Figure 2. Therefore, for � 2 (�G; 1]; the stricter the

patentability requirement, that is, the higher that IPG is raised above ImG ; the more urgent it

becomes to increase the emission tax simultaneously in order to curb emissions.

4.4 Socially Optimal Investment

The socially optimal level of investment, given that the �rm sets the monopoly price in the

second period, is the solution of

MaxIG

Wm(IG) = �m(IG)� IG + CSm(IG) + �e(IG)� CPG; (15)

and is denoted by I�G.29 The consumer surplus is given by:

CSm(IG) =1

2�m(IG)� e(IG):

Thus, I�G must satisfy dW (IG)=dIG = 0 or, equivalently,

d�m(IG)dIG

� 1 + dCSm(IG)dIG

+ � de(IG)dIG= 0: (16)

29We assume that taxes are a redistribution between consumers and producers. That is, taxes reduce pro�ts

and are re-distributed in a lump sum way to consumers. Since �m(IG); as de�ned by (10), denotes pro�ts net of

taxes, we add the tax revenue, �e(IG); back in the social welfare function, given by (15).

25

Evaluated at ImG , the left-hand side of (16) above becomes

1

2+ (� � 1)de(IG)dIG

: (17)

By Lemma 1, we have that dem(IG)=dIG < (>) 0 for � < (>)�G. Thus, the expression (17) is

positive for � > �G and � > 1 and for � < �G and � < 1: The expression (17) may be negative

for � > �G and � < 1 and for � < �G and � > 1: There exists a threshold e�1 > 1 such that for� < �G and � > e�1; we have (17) < 0: There exists a threshold e�2 < 1 such that for � > �G and� < e�2; we have (17) < 0: This leads to the following Proposition.Proposition 4 (i) For � 2 (�1; �G] and � � e�1; ImG < I�G (the �rm underinvests relative to

the socially optimal level) and for � > e�1, ImG > I�G (the �rm overinvests relative to the

socially optimal level).

(ii) For � 2 (�G; 1] and � � e�2; ImG < I�G (the �rm underinvests relative to the socially optimal

level) and for � < e�2, ImG > I�G (the �rm overinvests relative to the socially optimal level).

Proposition 4(i) states that as long as the fraction of green conscious consumers is su¢ ciently

low, the �rm underinvests relative to the socially optimal level for a su¢ ciently low emission tax

and overinvests for a su¢ ciently high emission tax. This is in line with much of the literature

surrounding the Porter hypothesis which predicts that higher taxes induce more investment by

�rms. However, the gap between the socially e¢ cient and privately optimal levels of investment

steadily reduces as � increases, until this result is reversed when the fraction of green conscious

consumers is su¢ ciently high, as stated by Proposition 4(ii).

5 Duopoly

We have thus far assumed that once a patent has been granted for a green technology that

reduces the emission-output ratio below the threshold P , the patented technology becomes

a new technology standard for the industry which is enforced by regulators. This prevents

other �rms from producing the good with the old dirty technology, and allows us to focus on

the monopoly case in order to clearly isolate and identify the direct channel through which

26

consumers�behaviour impacts innovation and the emission level. In this section, we relax this

assumption and consider a scenario where the new green technology can coexist with the old

dirty technology in a duopoly game. We show that the paradoxical result whereby an increase

in emission tax increases the emission level still holds for a range of parameter values even after

allowing �rms to behave strategically when setting prices.

In this section, we assume that Firm 1 produces a green product with an emission-output

ratio given by G and price given by pG: Firm 2 produces a dirty product with an emission-

output ratio given by H and price given by pH ; where H > G and pH < pG. That is, we

allow Firm 1 to unilaterally invest in reducing . While all the details of the derivations of this

duopoly model are provided in Appendix B, in this section, we summarize our main �ndings

within this duopoly setting.

A green conscious consumer has the following utility function:

UG =

8>>>><>>>>:v � GG� pG � P (e) from buying the product from Firm 1

v � HG� pH � P (e) from buying the product from Firm 2

�P (e) from not buying

(18)

A non-green conscious consumer always buys the dirty product since it is cheaper, and has the

following utility function:

UNG =

8<: v � pH � P (e) from buying the product

�P (e) from not buying(19)

A green conscious consumer G buys the green product rather than the dirty product as long

as the following condition holds:

v � GG� pG � P (e) > v � HG� pH � P (e);

that is, as long as G 2�pG�pH H� G

; Gi:Also, a green conscious consumer buys the green product

rather than not buying anything as long as v � GG � pG � P (e) > �P (e); that is, G <

(pG � v)= G:A non-green conscious consumer buys the dirty good as long as v � pH .

27

When both �rms face positive demands, their demand functions are given by:30

DH = �(pG � pH H � G

�G) + (1� �);

DG = �(v � pG G

� pG � pH H � G

):

Within this setting, the impact of increasing investment by Firm 1 on the emission level is

illustrated by Figure 5.

Figure 5: Impact of Increasing Investment in Green Technology on Emissions on Duopoly Case

Similarly to Figure 2, Figure 5 has an area (top area where � > maxf�; �3( G)g) where

the emission level increases in the investment level in green innovation (@e=@IG > 0). As the

investment level increases, the emission of Firm 2 is reduced (as more consumers buy the cleaner

product), but the emission of Firm 1 increases more than the decrease in the emission level of

Firm 2. Thus, we illustrate that the key driving force behind the main results under monopoly

carries over to the case of duopoly where a �rm can strategically choose the greenness of its

product.

Moreover, our results in terms of emission levels are qualitatively similar to those summarized

by Figures 3 and 4 as long as H is su¢ ciently large. First, in the case of Bertrand competition

(if Firm 1 does not innovate, and both �rms produce the dirty product), we obtain the same

equilibrium as in the monopoly case: the emission level is given by ec = �(v� c� H(G+ �)) +

H(1��); which is decreasing with � . We de�ne b� to be the value of the tax for which ec = 0; as30 In Appendix B, we provide the conditions on prices pG and pH such that both �rms face positive demands.

28

before. Then, we consider the case when the patenting constraint binds and Firm 1 must invest

IPG(> IdG) in order to patent its innovation. In this case, the emission levels are, thus, given

by eG(IPG) = PDG(�) and eH(IPG) = HDH(�), and both of them are decreasing with the

tax, � ; as @eG=@� = P@DG=@� < 0 and @eH=@� = H@DH=@� < 0. Finally, we show that in

the case where the patenting constraint is non-binding, as long as H > 2 G; the total emission

level is decreasing with the tax, @e(�)=@� < 0. Evaluated at � = 0, e(IdG) < ec, so that the

emission levels in the case of our duopoly setting are similar to the monopoly case as represented

in Figures 3 and 4. In particular, we retrieve the paradoxical result where an increase in � leads

to a increase in emission level for � 2 (�2; �3). However, in the duopoly case, whether �3 is less

that b� is ambiguous and depends on the values of P and H .6 Conclusion

In this paper, we develop a theoretical framework to investigate the impact of patent policies

and emission taxes on green innovation and emissions in the presence of environmentally friendly

consumers. We analyze the e¤ect of changing patentability requirements and patenting costs

when a �rm may invest in a green innovation, which reduces the emission output ratio.

We show that, in the absence of green consumers, the introduction of patents may result in

a paradoxical result whereby increasing the emission tax beyond a certain threshold leads to a

discrete increase in the emission level, which may be avoided by reducing the patenting cost, e.g.,

by fast-tracking green patents. In the presence of green consumers, this paradox is restricted

to an intermediate range of tax rates. This is because, at very high tax rates, the emission

level in the competitive equilibrium that occurs in the absence of innovation, is lower than that

with innovation. Thus, a lower emission level is reached despite a tax increase that makes the

innovation unpro�table. It follows that, at su¢ ciently high tax rates, reducing patenting costs

in order to induce innovation ends up increasing the emission level in the presence of green

consumers.

Moreover, we show that a stricter patentability requirement is only e¤ective at reducing

emissions as long as the fraction of green consumers is su¢ ciently small. We also �nd that

29

investment in green technologies reduces emissions only if the fraction of green consumers is suf-

�ciently small, and that the magnitude of this e¤ect decreases as the fraction of green consumers

increases. If the fraction of green consumers increases beyond a certain threshold, investment in

green technologies results in more emissions. To prevent this perverse outcome, policy makers

might consider imposing a capacity constraint on �rms as they invest more in green innovation

in the presence of green consumers. For a su¢ ciently small fraction of green consumers, we

retrieve the expected result found in much of the literature surrounding the Porter hypothesis

that a higher emission tax increases green investment. However, this result is reversed if the

fraction of green consumers rises to a level such that investment in green technologies results in

more emissions. Finally, we show that as long as the fraction of green consumers is su¢ ciently

low, the �rm underinvests relative to the socially optimal level for a su¢ ciently low emission

tax and overinvests for a su¢ ciently high emission tax. However, the gap between the socially

e¢ cient and privately optimal levels of investment steadily reduces as this fraction increases,

until this result is reversed when the fraction of green consumers is su¢ ciently high.

To summarize, this paper determined the conditions under which reducing patenting costs

and making patentability requirements stricter are e¤ective at inducing green innovation and

thereby reducing emissions. These patent policy tools were shown to become less e¤ective

as society becomes more environmentally friendly. Thus, further research seems warranted

regarding the policies to reduce emissions through green innovation, and also regarding the

type of information to distribute to consumers. In particular, if consumers reward marginal

reductions in emission-ouput ratios of production processes by increasing their demand for the

�nal product, this may inhibit the traditional patent policy tools from working as expected. At

the same time, increasing emission taxes to very high levels seem more e¤ective in the presence

of green consumers, while increasing emission taxes may be harmful for the environment in

the absence of green consumers. These are some of the policy implications generated in our

framework.

As implied by the above summary of our �ndings, this paper generates a number of po-

tentially testable hypotheses, each of which depends on the fraction of green consumers, �:

There are a few indices that measure the environmental friendliness of consumers across coun-

30

tries, such as Greendex (computed by National Geographic) and GlobeScan,31 which may be

used in conjunction with data on the relevant variables, such as emission and green investment

levels and patenting costs, to test the hypotheses generated by the Lemmas and Propositions

in this paper. Instead of using green consumer indices, another possible approach to testing

our hypotheses might be the following two step procedure. First, the �rebound-like� e¤ect as

explained in Footnote 9, whereby the demand of a product increases the cleaner it becomes,

would have to be estimated for di¤erent goods/regions. This would then act as a proxy for �:

Second, the interaction of this �rebound-like� e¤ect with the relevant variables would need to

be estimated. Testing the above hypotheses is beyond the scope of this paper, and we leave the

relevant empirical analysis as future research.

The �ndings of this paper give rise to a number of questions which would be interesting to

address in future work on this topic by extending the framework developed here. For example,

in an open economy setting, where the degree of environmental friendliness of consumers is het-

erogeneous across countries, and polluting �rms with market power are located across di¤erent

countries, how are governments�strategies regarding the implementation of Intellectual Property

Rights and fast-track patent systems a¤ected? How do these policies a¤ect the distribution of

investment, pollution and welfare levels across countries? Also, in a general equilibrium setting,

how does a drop in demand in the sector under consideration a¤ect consumption in other sectors,

and how does this a¤ect total emissions?

31For further details refer to https://globescan.com/greendex-2014-consumer-choice-and-the-environment-a-

worldwide-tracking-survey-full-report/.

31

Appendix A

Figures

Figure A1

Figure A2

Proof of Proposition 1

The First Order Condition (FOC) in the �rst stage is given by:

�� 0�ING

�� 1 = 0: (20)

32

Let f�� ; ING

�� �� 0

�ING

�� 1. We then totally di¤erentiate FOC (20), f�d� + fIdING = 0.

Thus,dING

d�= �f�

fI= � � 0(ING)

�� 00(ING) > 0:

Moreover,@e

@�=@

@I

@ING

@�< 0:

When � = 0; we have ING = 0 with = H ; with decreasing in � for ING > 0: At � = �̂1;

we have �ING

�= P : Thus, for a su¢ ciently low tax rate, that is, � 2 [0; �̂1); the �rm invests

the minimum amount required to obtain a patent, that is, P . For �1 2 [�̂1; �̂2), the �rm has

an incentive to invest more than this minimum amount since �ING

�< P . We note that the

�rm only invests as long as

v � c� � �ING

�� ING � CPG > 0; (21)

which implies that

v � c� CPG > � �ING

�+ ING;

with @@�

�� �ING

�+ ING

�=

�ING

�+ @I

@�

�� @ @I + 1

�: By (20) ; we have that

�� @ @I + 1

�= 0

such that @@�

�� �ING

�+ ING

�> 0: Thus, there exists a threshold �̂2 such that for � 2 [0; �̂2);

we have v� c�CPG > � �ING

�+ ING and for � > �̂2; we have v� c�CPG � �

�ING

�+ ING:

It follows that for � > �̂2, the �rm does not invest and Bertrand competition results in emission

level given by ec = H : �

33

Monopoly Pricing

The �rm chooses pm as de�ned by (8) instead of v if the pro�t (10) is higher than the pro�t

(7), i.e., �m > �(v) or, equivalently,

�

4 [v � c� (G+ �) + 1��

� ]2 > (v � c� �)(1� �);

[v � c� � � G+ 1��� ]

2 > 4 1� ��

(v � c� �):

Let us denote a = v�c� � and b = 1��� . The inequality can be rewritten as [a� G+b]

2 > 4ba

or

b2 � 2b(a+ G) + (a� G)2 > 0: (22)

Thus, the discriminant of the quadratic equation is � = 4(a+ G)2� 4(a� G)2 = 16a G > 0,

and the two roots are

b0 = (a+ G)� 2pa G > 0;

b00 = (a+ G) + 2pa G:

Therefore, for values of b such that b < b0 and b > b00 inequality (22) is satis�ed. In terms of �;

we obtain that this inequality is satis�ed as long as � > �1 and � < �2 with

�1 =

+v�c+ (G��)�2(v�c� �)12 ( G)

12;

�2 =

+v�c+ (G��)+2(v�c� �)12 ( G)

12:

We show that �2 < �, so that the �rm will choose the monopoly price (8) if � > �1.

Proof of Lemma 1

From (12), we have @em(IG)@� = ��

2 < 0; which results in Lemma 1(i): The derivative of em(IG)

with respect to IG is given by:

@em(IG)@IG

=1

2(1� � (G+ � + 1)) d dIG ;

where d =dIG < 0. Thus, @em(IG)=@IG < (>) 0 for � < (>)�G; which results in Lemma 1(ii).

�

34

Second Order Condition

In the �rst period, the �rm chooses the optimal investment that satis�es�MaxIG

�m(IG)� IGs.t. (IG) � P

where the pro�t �m(IG) is de�ned by (10). Let ImG be the solution of the unconstrained opti-

mization program (i.e., ImG = argmax�m(IG)� IG) such that it is solution of (13). Notice that

the expression

�4 ((

v�c )

2 � (G+ �)2) + (1��)2 (G+ �)� 1

4(1��)2�

is strictly concave for all 0 < � < 1; with two roots given by ~� = v�c+ +G +� and

�� =

c�v+ +G +� > 1: It can be shown that

~� < � < �G < 1 < ��. Therefore, the left-hand side of

equation (13) is positive such that (13) is satis�ed.

The Second Order Condition (SOC) is satis�ed as long as

�

2

� 0�2 (v�c)2

3� 00(�4 ((

v�c )

2 � (G+ �)2) + (1� �)(G+ �)� 14(1��)2� ) < 0:

Using the First Order Condition (13) we can write

�(�4 ((v�c )

2 � (G+ �)2) + (1� �)(G+ �)� 14(1��)2� ) = 1

0(IG);

that we plug into the SOC to obtain a local condition

�

2

� 0�2 (v�c)2

3+ 00

0(IG)< 0;

or

00(:) > �� (v�c)2

2 ( 0(:) (:) )

3;

which is Assumption (A3). If this local condition is satis�ed, locally the pro�t function is

concave.

Comparative Statics

Lemma 5 For a weak patentability requirement, ImG is decreasing in c, and increasing in v.

Lemma 5 follows directly from (13) (see below for details of the calculations). As the marginal

cost of production, c; increases, the pro�tability of the product decreases, which explains why

35

�rms have less incentive to invest in the green technology. As v increases, demand increases,

leading to an increase in ImG :

Proofs of Lemmas 2 and 5

In order to derive the relevant comparative statics results on ImG ; it is useful to de�ne the

following:

F (IG; y) � � 0(�4 ((v�c )

2 � (G+ �)2) + (1� �)2

(G+ �)� 14(1��)2� )� 1;

where y may represent any of the exogenously given parameters, that is, y 2 f� ; c; v;G; �g. By

totally di¤erentiating (13), we have the following

dImGdy

= �@F@y

@F@IG

:

Since we have assumed the existence of an interior solution, it follows that a maximum is reached

at ImG : This implies that@F@IG

� 0, and thus

signdImGdy

= sign@F

@y:

For y 2 f� ; c; v;G; �g, we have

@F

@�=@F

@G= 0

2(�(� +G)� (1� �)) < 0;

if and only if � � �G.

@F

@c= �

0

2

v � c 2

< 0;

@F

@v= ��

0

2

v � c 2

> 0:

The following table summarizes the above comparative statics results.

TABLE 1: Comparative Statics

Parameter y sign�dImGdy

�� -

c -

v +

G -

36

Figure 3

The emission level when the price is equal to marginal cost is given by equation (14), ec =

�(v � c� H(G+ �)) + H(1� �) which is decreasing with � at the constant rate �� H . The

emission level is positive for � < b� � (v � c)= H � G + (1 � �)=�, which is satis�ed accordingto Assumption (A1). It is represented by the top (red) function. The bottom (green) function

represents the emission level when the patenting constraint is binding em(IPG) = �(v�c� P (G+

�))=2+(1��) P =2: This function is also decreasing with � at the constant rate �� P =2 > �� H .

The emission level is positive for � < (v�c)= P�G+(1��)=� where (v�c)= P�G+(1��)=� > b� .The two functions intersect at � = (v � c)=(2 H � P ) � G + (1 � �)=�. The middle (blue)

curve represents the emission level at the pro�t-maximizing investment level ImG , and it intersects

with em(IPG) at e� which is where (ImG ) = P or ImG = IPG. We also show that, evaluated at

� = 0, em(IPG) < ec is satis�ed if H < (v � c)=2G and we have em(ImG ) > em(IPG) as long as

� > 1=(G+ 1).

Proof of Lemma 4

As long as IPG � ImG , i.e., the patentability requirement is weak, and CPG < �m(ImG ) � ImG ; it

is pro�table to invest. If IPG > ImG for all � , i.e., the patentability requirement is strong, then

unless CPG < �m(IPG)� IPG < �m(ImG )� ImG ; it is not pro�table to invest. �

Proof of Proposition 2

The emission level, em(Im); is decreasing in � since

@em(Im)

@�=dem

d�+@em(Im)

@Im@Im

@�< 0: (23)

For any �2 > ~� ; the relevant emission level is em(IPG) since we have that IPG > ImG for this range

of � . If Im = IPG, the second term of (23) is null, and we have that dem=d� = �� (Im)=2 < 0.

For any �1 < ~� ; the relevant emission level is em(ImG ) since we have that IPG � ImG for this

range of � . If Im = ImG , the second term of (23) is also negative since @em=@Im < (>) 0 for

� < (>)�G by Lemma 1; and @Im=@� > (<) 0 for � < (>)�G by Lemma 2. It follows that���@em(ImG )@�

��� > ���@em(IPG)@�

��� : This completes the proof of Proposition 3(i). An increase in � fromeither �1 or �2 to �3 causes �m(ImG )� ImG to fall below CPG such that it becomes unpro�table

37

for the �rm to invest in the green technology, as per Proposition 2. This causes the emission

level to rise from em(Im) to ec as given by (14) : This completes the proof of Proposition 3(ii).

�

Socially Optimal Investment

The total welfare is

Wm(IG) = �m(IG)� IG � CPG + �e+ CSm(IG);

where �m(IG) is de�ned by (10) and the consumer surplus is

CSm(IG) = �

Z v�p

G(v � G� p� P (e))dF + �

Z G

v�p

(�P (e))dF + (1� �)(�P (e))

= �8 (v � c� (G+ �)�

1��� )

2 � e:

Therefore, the total welfare can be written as

Wm(IG) =3

2(�m(IG)� IG) +

1

2IG � (1� �)e� CPG:

The derivative of the total welfare gives

dWm(IG)dIG

=3

2(d�

m(IG)dIG

� 1) + 12� (1� �)de(:)dIG

:

Evaluated at ImG , it becomes expression (14) or

1

2+ (� � 1)de(:)dIG

:

It is positive for � > �G and � > 1 and for � < �G and � < 1: There exists a threshold e�1 > 1such that for � < �G and � > e�1; we have (17) < 0: There exists a threshold e�2 < 1 such thatfor � > �G and � < e�2; we have (17) < 0:

38

Appendix B �Duopoly Case

Consider a duopoly case in which one of the �rms, Firm 1, gets a patent for the green technology

it has discovered, while the other �rm, Firm 2, still produces its dirty product. Firm 1 produces

a green product, the emission-output ratio of which is given by (IG) = G at price pG, and

Firm 2 produces the dirty product, the emission-output ratio of which is given by H at price

pH ; where G < H . Let�s assume pG > pH .

The timing is similar to the monopoly case: in the �rst stage of the game, Firm 1 decides the

level of investment in the green technology, IG. Once the innovation has been discovered, Firm 1

decides whether to patent it. In the second stage, both �rms choose their prices simultaneously.

The patent prevents Firm 2 from producing with the cleaner technology and, thus, Firm 2 can

only use the dirty technology.

Unlike the monopoly case, there are now two demand functions: one for the green product

and one for the dirty product. A green conscious consumer has the following utility function:

UG =

8>>>><>>>>:v � GG� pG � P (e) from buying the product from Firm 1

v � HG� pH � P (e) from buying the product from Firm 2

�P (e) from not buying

A non-green conscious consumer has the following utility function:

UNG =

8<: v � pH � P (e) from buying the product

�P (e) from not buying

A green conscious consumer G buys the green product rather than the dirty product as long

as v � GG � pG � P (e) > v � HG � pH � P (e); that is, as long as G 2 ( pG�pH H� G; G]:Also, a

green conscious consumer buys the green product rather than not buying anything as long as

v� GG� pG � P (e) > �P (e); that is, G < (pG � v)= G:A non-green consumer buys the dirty

good as long as v � pH .

39

The demand for the dirty product is, thus, given by:

DH =

8>>>>>>><>>>>>>>:

1 if pH � v < G HpH + v

H� G H

< pG

�( pG�pH H� G�G) + (1� �) if pH � v < pH + ( H � G)G < pG <

G HpH + v

H� G H

(1� �) if pH � v < pG < pH + ( H � G)G

0 if v < pH < pG

The demand for the green product is given by:

DG =

8>>>><>>>>:� if pH � v < pG < pH + ( H � G)G

�(v�pG G� pG�pH

H� G) if pH � v < pH + ( H � G)G < pG <

G HpH + v

H� G H

0 if pH � v < G HpH + v

H� G H

< pG

Green Conscious Consumers �Duopoly Pricing

Given that Firm 1 has made an innovation and has patented it, we determine the equilibrium

prices for a given investment level IG.

First, we show that Firm 2 will not set its price pH = v as the condition v < pG will not

be satis�ed. Thus, Firm 2 must set its price at pH < v. When the �rms serve both types of

consumers, Firm 1 will choose pG that solves the following:

Max(pG � c� � G)�(v

G+

pH H � G

� pG H

G( H � G)):

We obtain the best response function of Firm 1 as a function of the price of the dirty product,

as follows:

pG(pH) =1

2( G HpH +

v

H( H � G) + c+ � G):

Firm 2 chooses pH that solves the following:

Max(pH � c� � H)(�(pG � pH H � G

�G) + (1� �));

which gives the following best response function:

pH(pG) =1

2(pG + (

1� ��

�G)( H � G) + c+ � H):

Solving the two best response functions simultaneously, we obtain the following pro�t-maximizing

prices:

p�G = 14 H� G

(( H � G)( G(1��� �G) + 2v) + ( G + 2 H)c+ 3� H G);

p�H = 14 H� G

(( H � G)(2 H(1��� �G) + v) + ( G + 2 H)� H + 3 Hc):

40

We verify that p�G > p�H if v � c � 2� H + G(2 H � G) > 0. We also check that p�H < v if

� > �1( G); where

�1( G) �2( H� G)

3(v�c� G�)+(2G+1�2�)( H� G):

To obtain positive demands for both products, we check that v�( H� G)G < p�H if � < �2( G);

where

�2( G) �2( H� G)

(v�c)+(2 H+ G)(G��)+2( H� G)+2 GG+ G G H

G);

and that ( H � G)G < p�G � p�H if � > �3( G); where

�3( G) �2 H� G

v�c�2 H(�+G)+2 H� G:

Lastly, we need to check that p�G < G Hp�H + v

H� G H

since � < 1.

Both demands evaluated at prices (p�G; p�H) are, thus, given by:

DH(p�G; p

�H) =

14 H� G

(�(v � c� 2 H(� + 1 +G)) + 2 H); (24)

and

DG(p�G; p

�H) =

H G(4 H� G)

(�(2(v � c)� G(G+ � + 1)) + G): (25)

The demand for the dirty product decreases with the investment level in green technology, IG;

as

@DH(IG)@IG

= � 0G(IG)v�c�2(G+�� 1��

�) H

(4 H� G)2< 0:

However, the demand for the green good decreases with the investment level, IG; only if � <

�5( G), but increases with the investment level for � > �5( G); where

�5( G) � 2G

2G(G+�+1)+4(v�c)(2 H� G);

where �5( G) < �4(IG): The response of demand to marginal changes in investment is given by:

@DG(IG)@IG

= H 0G(IG)

2G�� 2G(G+�+1)�4�(v�c)(2 H� G) 2G(4 H� G)2

:

If the fraction of green conscious consumers, �, is large enough, an increase in the investment

level in green technology increases the demand for the green product.

41

The emission from the dirty producer is given by eH = HDH(p�G; p

�H), which is always

decreasing with the investment level, IG; as

@eH@IG

= H� 0G(IG)

v�c�2(G+�� 1���) H

(4 H� G(IG))2< 0:

As Firm 1 invests in cleaner technology, the emission of the dirty product will be reduced as the

demand decreases.

The emission from the green producer is given by eG = G(IG)DG(IG): As the investment

level increases, the emission level increases if � > � where

� � 2 H2(G+�+1) H�(v�c)

;

as@eG@IG

= 0G(IG) H2

(4 H � G)2(�(v � c)� �(G+ � + 1)2 H + 2 H):

We can also derive the total emission level given by e = eH + eG = HDH + G(I)DG(I), which

is decreasing in � as@e

@�= H

��(2 H + G(I))4 H � G(I)

< 0:

For a given investment level, as � increases, the total emission level decreases. However, when

we account for the pro�t-maximizing investment level, the e¤ect of an increase of investment on

the emission level can be decomposed as follows:

@e

@IG=@eH@IG|{z}(�)

+@eG@IG|{z}

(+ or �)

:

Thus, an increase in the investment level decreases the emission level of Firm 2, but may increase

or decrease the emission level of Firm 1. Formally, we have that

@e

@IG= 0G

3 H(4 H � G)2

(�((v � c)� 2(G+ � + 1) H) + 2 H);

so that @e=@IG > 0 if � > �. We summarize these �ndings in the following Figure, which is the

same as Figure 5 in the paper.

Similarly to Figure 2, Figure 5 has an area (top area where � > maxf�; �3( G)g) where

the emission level increases in the investment level in green innovation (@e=@IG > 0). As the

42

investment level increases, the emission of Firm 2 is reduced (as more consumers buy the cleaner

product), but the emission of Firm 1 increases more than the decrease in the emission level of

Firm 2.

Pro�t-Maximizing Investment

At the outset, Firm 1 will choose IG that solves the following problem:

�MaxfIG

1� H( H � G(IG))

(2�(v�c)�� G(v)(G+�+1)+ G(IG))2 G(IG)(4 H� G(IG))2

� IGg

s.t. (IG) � P � (IPG)

Let ID(�) denote the solution of this maximization problem, which is either IPG (the mini-

mum investment needed to obtain a patent) or IdG (if it is larger than IPG). Note that IdG is

the unconstrained pro�t-maximizing investment level, which satis�es the following �rst order

condition:

� 0G(:) H��((v�c)(4 2G�6 G H+8 2H)+ G H(1+�+G)(4 H�7 G))+ G H(4 H�7 G)

� 2G(4 H� G)3 (�( G(G+�+1)�2(v�c))� G) = 1:

(26)

Let F (IdG; �) represent this �rst order condition, (26). The solution IdG is a function of � and,

thus, by totally di¤erentiating F (:) and rearranging the terms, we can obtain the sign of @IdG=@� ;

which is the same as the sign of @F=@� , when the SOC condition is satis�ed. We, thus, calculate

the following:

@F

@�= 2( H(4 H�7 G)��( H(4 H�7 G)(G+�+1)+2(v�c)( G+2 H)))

(4 H� G)3;

which is negative if � > e� wheree� � H(4 H�7 G)

H(4 H�7 G)(G+�+1)+2(v�c)( G+2 H):

Therefore, if the fraction of green conscious consumers is large enough, � > e�, as the taxincreases the equilibrium investment level decreases, @IdG=@� < 0.

Emission levels in equilibrium

First, in the case of Bertrand competition (if Firm 1 does not innovate, and both �rms produce

the dirty product), we obtain the same equilibrium as in the monopoly case: the emission level

43

is given by ec = �(v � c� H(G+ �)) + H(1� �); which is decreasing with � . We de�ne b� tobe the value of the tax for which ec = 0,

b� = v � c H

�G+ 1� ��

:

Then, we consider the case when the patenting constraint binds and Firm 1 must invest

IPG(> IdG) in order to patent its innovation. In this case, the emission levels are, thus, given by

eG(IPG) = PDG(�) and eH(IPG) = HDH(�), and both of them are decreasing with the tax,

� ; as @eG(�)=@� = P@DG=@� < 0 and @eH=@� = H@DH=@� < 0. Thus, the total emission

level is given by

ePG = H

4 H� P(3�(v � c)� �(G+ � + 1)( P + 2 H) + 2 H + P );

which is decreasing with the tax as

@ePG@� = �� H

2 H+ P4 H� P

< 0:

We show that ePG intersects ec at

�3 =v � c2 H

�G+ 1� ��

;

where �3 < b� .Evaluated at � = 0, the total emission level is ePG(� = 0) = PDG(0) + HDH(0), which is

smaller than ec(� = 0) if �(v � c)� 2 H�(G+ 1) + 2 H > 0:

When the patenting constraint does not bind, Firm 1 chooses I�G that satis�es (26). In that

case, the equilibrium emission level as a function of � is given by:

e(�) = eH(�) + eG(�) = HDH(IdG(�); �) + G(I

dG(�))DG(I

dG(�); �):

Note that the demands are de�ned by (25) and (24) when Firm 1 chooses the pro�t-maximizing

level of investment IdG, which means that IdG is higher than the minimum requirement to obtain

a patent IPG.

First, we consider the impact of � on the emission level of Firm 2, eH(�);

@eH@�

= H(@DH@IG

@IdG@�

+@DH@�

);

44

where @DH=@� < 0 and @DH=@IG < 0. Thus, if @IdG=@� > 0, which occurs when the fraction of

green conscious consumers is not large, � < e�, then an increase in the tax reduces the emissionlevel of Firm 2, @eH=@� < 0. However, if the fraction of green conscious consumers is large

enough, � > e�, we have @IdG=@� < 0, which implies that the impact of an increase of the tax onthe emission level is ambiguous. If we calculate @DH

@IG

@IdG@� +

@DH@� ; we obtain the following:

2 0G(IG)Id0G (�)

�(v�c)�2 H�(G+�+1)+2 H(4 H� G)2

� 2� H(4 H� G)(4 H� G)2

;

which is negative if

� <2 0G(IG)I

d0G (�) H

�( 0G(IG)Id0G (�)(v�c�2 H(G+�+1))� H(4 H� G))

;

which is higher than 1 for 0G(IG)Id0G (�)(v � c� 2 H(G+ �)) > H(4 H � G).

The e¤ect of an increase of the tax on the emission level of Firm 1, eG(�) = G(IdG(�))DG(I

dG(�); �),

is determined by

@eG(�)

@�=

@ G@IG

@IdG@�

DG(:) + G(I�G)(

@DG@IG

@IdG@�

+@DG@�

);

=@IdG@�|{z}

(� or +)

(@ G@IG|{z}(�)

DG(:) + G(IdG)

@DG@IG| {z }

(� or +)

) + G(IdG)@DG@�| {z }(�)

:

Thus, as @DG=@� < 0, if @DG=@IG < 0 (if � < �5( G)) and @IdG=@� > 0 (for � < e�) then

@eG(�)=@� < 0. We also calculate that e� > �5( G) if (4 H � G) � 2G + 4 H( H � 2 G�) > 0,or if H > 2 G. Therefore, as long as H > 2 G the total emission level is decreasing with the

tax, @e(�)=@� < 0. Evaluated at � = 0, e(I) < ec, so that the emission level in the case of a

duopoly is similar to the monopoly case as represented in Figures 3 and 4.

45